Armature not completely moving head mesh
I noticed that marios right arm gets twisted when the armature gets parented.
To fix that, you need to go into pose mode of the armature, press CTRL + A and select apply pose as rest pose. 

Even though the eyes,mustache, and face are one object, the eyes/mustache arent connected to the face but floating in mid air. So the automatic weights didnt work perfectly. You can see that because when in weight paint mode, the eyes are blue (meaning the vertices arent assigned to that bone) and the mustache is green (meaning it is only partially controlled by that bone)

You can easily fix that by manually selecting all the vertices in the face with A  and assigning all of them to be fully controlled by the head bone in the Properties Editor->Data->Vertex Groups->Assign. I also noticed the mustache is partially controlled by the chest bone so I removed it from its control.

The process is called "weight painting".